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Maxwell's equations (Ampere's law and Faraday's Law) say that:

$$\nabla \times \vec{E} = - \frac{\partial B}{\partial t}$$

To me, the derivative on the LHS is a spatial one, the partial on the RHS is a time varying one. So all we can deduce from this is that a time varying magnetic field generates a curly E field (says nothing about time varying E field). So why do textbooks typically make the statement that time varying E field produces a time varying B field ?

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There is

$$ \nabla \times \vec{B} = \mu_0 \big( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \big) $$

as well.

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  • $\begingroup$ Yes, but this is saying a time varying E field generates a curly magnetic field. We don't know if the time varying magnetic field is necessary curly, it could just be time varying, does it not generate an E field then ? $\endgroup$
    – Andrea Escalante
    Feb 12, 2018 at 3:23
  • $\begingroup$ Try differentiating this equation with $t$. See if/how to get a $\vec{B}$ that is curly but not time dependent. $\endgroup$
    – AHusain
    Feb 12, 2018 at 3:28

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